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Krus, D. J. (2001) Matrix subtraction. Journal of Visual Statistics at www.visualstatistics.net (February 8).

 

Matrix subtraction

David J. Krus

Arizona State University

The operations on matrices differ from similar operations of scalar algebra in several respects. The matrix algebra operations, in general, are not commutative and attention must be paid to whether the matrices are conformable with respect to the intended operation. Also, it must be noted whether the matrix operation pertains to matrix elements or to matrices.

Subtraction of matrix elements

To subtract elements of matrices A and B,

    C = A (-) B

elements of matrix B are subtracted from their corresponding elements in matrix A and stored as elements of matrix C. Obviously, all three matrices must have the same dimensions (the same number of rows and columns). Notice that the minus sign in the above equation is enclosed in parentheses to indicate subtraction of matrix elements, as contrasted with subtraction of matrices. The operation of subtracting matrix elements can be schematized as
 

 

For instance

 

Subtraction of matrices

Textbooks on matrix algebra (cf., Horst, 1963; Shores, 2003), routinely describing major and minor vector products, do not include analogical operations for the major and minor differences of minuends and subtrahends. These operations are easy to imagine but not discussed, as most of their potential applications can be as well accomplished by other matrix algebra operations. However, on close scrutiny, the matrix algebra operation of subtraction (of vectors, not elements of vectors, of matrices, not elements of matrices) can be used for concise expression of many abstract concepts within the matrix algebra framework. To subtract matrices ''A'' and ''B'',

  

the number of columns in matrix ''A'' must equal the number of rows in matrix ''B'', in another words, the matrices must be conformable to matrix subtraction. The resulting matrix ''C'' will have the number of rows of the first matrix and the number of columns of the second matrix. The schematic representation of matrix subtraction is shown below

 

For instance

Skew-symmetric matrices

Subtracting a transpose of a matrix (or a vector) from itself yields a skew-symmetric matrix. For instance,

             

Skew-asymmetric matrices

Skew-symmetric matrices can be transformed into skew-asymmetric matrices by asymmetrization, i.e., by deleting all negative elements. Thus, e.g., the above skew-symmetric matrix can be transformed to a skew-asymmetric matrix as

 

Dendrograms

The operation of matrix subtraction facilitates expression of many abstract concepts within the matrix algebra framework. For example, the true variance of the variable X [0, 1, 2, 3] is computed by subtracting the mean (1.5) from the variable X, thus transforming the vector of the obtained scores X into the vector of the deviation scores x = [-1.5, -.5, .5, 1.5]. Squaring the values of the deviation scores x as [2.25, .25, .25, 2.25], and by averaging these squared values as (5/4), equals 1.25, the variance of the vector X.

 

Using the matrix subtraction with subsequent asymmetrization as described above, the variance of the vector X [0, 1, 2, 3] can be computed from the elements of its associated skew-asymmetric matrix by averaging their squared values, i.e., as (1² + 2² + 1² + 3² + 2² + 1² ) / 4²) which also equals 1.25. The advantage of computing variance from the skew asymmetric matrices is that these matrices are also adjacent to ordered graphs, as shown in Fig. 1.

 

Fig.1. Dendrogram adjacent to the skew asymmetric matrix
associated with the vector X [0, 1, 2, 3].

 

Binaries

Vector X [0, 1, 2, 3] can be transformed into its adjacent binary implicational matrix without a loss of information

 

since the matrix subtraction of the transpose of this binary matrix from itself

 

yields the same skew-symmetric matrix which can be obtained directly from the matrix subtraction of the transpose of the vector X from itself as

 

  

 

The fact that within the theory of information the information is tied into the number of the 0 - 1 differences between the values of the binary digits (bits) provides a link between the variance in the statistical analysis and information within the mathematical theory of information as formulated by Shannon and Weaver.  

 

Stochastic skew-asymmetric matrices

Generalizing the matrix algebra operations on subtracted data matrices as to envelop the '''stochastic skew-asymmetric matrices''', yields dendrograms such as shown in Fig. 2.


Fig. 2. Dendrogram associated with a stochastic skew
 asymmetric matrix of differences (Source).

These algorithms are among the basic matrix algebra operations used in visual statistics.


References

Horst, P. (1963). Matrix algebra for social scientists. New York: Holt.
Krus, D. J., & Ceuvorst, R. W. (1979). Dominance, information, and hierarchical scaling of variance space. Applied Psychological Measurement, 3, 515-527 (Request reprint).
Krus, D. J., & Wilkinson, S. M. (1986). Matrix differencing as a concise expression of variance. Educational and Psychological Measurement, 46, 179-183 (Request reprint).
Krus, D. J. (2002). Analysis of variance within the matrix algebra framework.

Shores, T. S. (2003). Applied linear algebra and matrix analysis. New York: McGraw-Hill.

See also

Matrix addition